no. 2
L p estimates for Schrödinger operators with certain potentials
Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 513-546.

Nous considérons des opérateurs de Schrödinger -Δ+V(x) dans ℝ n où le facteur non négatif V(x) appartient à la classe de Hölder inversée B q pour tout q≥n/2. Nous obtenons les estimations optimales L p pour les opérateurs (-Δ+V) iγ ,∇ 2 (-Δ+V) -1 ,∇(-Δ+V) -1/2 et ∇(-Δ+V) -1 où γ∈ℝ. En particulier nous montrons que (-Δ+V) iγ est un opérateur de Calderón-Zygmund si V∈B n/2 and ∇(-Δ+V) -1/2 ,∇(-Δ+V) -1 ∇ sont des opérateurs de Calderón-Zygmund si V∈B n .

We consider the Schrödinger operators -Δ+V(x) in ℝ n where the nonnegative potential V(x) belongs to the reverse Hölder class B q for some q≥n/2. We obtain the optimal L p estimates for the operators (-Δ+V) iγ ,∇ 2 (-Δ+V) -1 ,∇(-Δ+V) -1/2 and ∇(-Δ+V) -1 where γ∈ℝ. In particular we show that (-Δ+V) iγ is a Calderón-Zygmund operator if V∈B n/2 and ∇(-Δ+V) -1/2 ,∇(-Δ+V) -1 ∇ are Calderón-Zygmund operators if V∈B n .

@article{AIF_1995__45_2_513_0,
     author = {Shen, Zhongwei},
     title = {$L^p$ estimates for {Schr\"odinger} operators with certain potentials},
     journal = {Annales de l'Institut Fourier},
     pages = {513--546},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {2},
     year = {1995},
     doi = {10.5802/aif.1463},
     zbl = {0818.35021},
     mrnumber = {96h:35037},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1463/}
}
TY  - JOUR
AU  - Shen, Zhongwei
TI  - $L^p$ estimates for Schrödinger operators with certain potentials
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 513
EP  - 546
VL  - 45
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1463/
DO  - 10.5802/aif.1463
LA  - en
ID  - AIF_1995__45_2_513_0
ER  - 
%0 Journal Article
%A Shen, Zhongwei
%T $L^p$ estimates for Schrödinger operators with certain potentials
%J Annales de l'Institut Fourier
%D 1995
%P 513-546
%V 45
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1463/
%R 10.5802/aif.1463
%G en
%F AIF_1995__45_2_513_0
Shen, Zhongwei. $L^p$ estimates for Schrödinger operators with certain potentials. Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 513-546. doi : 10.5802/aif.1463. https://aif.centre-mersenne.org/articles/10.5802/aif.1463/

[CM] R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque, 57 (1978). | MR | Zbl

[D] Xuan Thinh Duong, H∞ Functional Calculus of Elliptic Partial Differential Operators in Lp Spaces, Ph.D. Thesis, Macquarie University, 1990.

[F] C. Fefferman, The Uncertainty Principle, Bull. Amer. Math. Soc., 9 (1983), 129-206. | MR | Zbl

[G] F. Gehring, The Lp-integrability of the Partial Derivatives of a Quasi-conformal Mapping, Acta Math., 130 (1973), 265-277. | MR | Zbl

[GT] D. Gilberg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Ed., Springer Verlag, 1983. | Zbl

[H] W. Hebish, A Multiplier Theorem for Schrödinger operators, Colloquium Math., LX/LXI (1990), 659-664. | Zbl

[HN] B. Helffer and J. Nourrigat, Une Ingégalité L2, preprint.

[M] B. Muckenhoupt, Weighted Norm Inequality for the Hardy Maximal Function, Trans. Amer. Math. Soc., 165 (1972), 207-226. | MR | Zbl

[RS] L. Rothschild and E. Stein, Hypoelliptic Differential Operators and Nilpotent Groups, Acta Math., 137 (1977), 247-320. | MR | Zbl

[Sh] Z. Shen, On the Neumann Problem for Schrödinger Operators in Lipschitz Domains, Indiana Univ. Math. J., 43(1) (1994), 143-176. | MR | Zbl

[Sm] H. F. Smith, Parametrix Construction for a Class of Subelliptic Differential Operators, Duke Math. J., (2)63 (1991), 343-354. | MR | Zbl

[St1] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. | MR | Zbl

[St2] E. Stein, Harmonic Analysis: Real-Variable Method, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. | Zbl

[T] S. Thangavelu, Riesz Transforms and the Wave Equation for the Hermite Operators, Comm. in P.D.E., (8)15 (1990), 1199-1215. | MR | Zbl

[Z] J. Zhong, Harmonic Analysis for Some Schrödinger Type Operators, Ph.D. Thesis, Princeton University, 1993.

Cité par Sources :